Eigenvectors of tensors and algorithms for Waring decomposition

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equations of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem).